Why does the "skin" modifier not accurately produce 90°corners?
I suggest using a combination of Screw + Solidify modifier (your base mesh must be flat for the best result) :

Screw : Along Z, 1 step (If using Blender  &lt; V2.83, the minimum steps is 2. Make sure the angle is 0°), Remove smooth shading

Solidify : Complex mode, offset 0



If you want to have the height centered around 0, simply add a displace with a driver :



Result :



Add custom properties if you don't want to deal with the modifier interface :